Characteristics and the three gap theorem. (English) Zbl 0709.11011

van Ravenstein, Tony; Winley, Graham; Tognetti, Keith

Fibonacci Q. 28, No. 3, 204-214 (1990).

Let [x] and \(\{\) \(x\}\) denote the integer and fractional parts of the positive real number x. If \(d_ m=[(m+1)x]-[mx]\) then \([mx]=\sum^{m- 1}_{i=1}d_ i+[x]\). Since each \(d_ i\) is [x] or \([x]+1\), we can define the characteristic of x to be the string \(x_ 1x_ 2x_ 3..\). where \(x_ i=s\) if \(d_ i=[x]\) and \(x_ i=\ell\) if \(d_ i=[x]+1\) (s for small and \(\ell\) for large). The authors exhibit a connection, for some x, between the characteristic of x and the sequence of arcs or gaps formed by partitioning a circle by successive placements of points by an angle of x revolutions. They do this by making use of the three gap theorem, which asserts that points placed in this way partition a circle into gaps of 2 or 3 different lengths.

Reviewer: Ian Anderson

MSC:

11B05	Density, gaps, topology
11A55	Continued fractions

Keywords:

greatest integer; fractional parts; three gap theorem